Thank you for posting. In the case of '~기', it indicates the 'act of doing something/process or method of the act''. You replace the '다' from the base form of a verb and replace it with '기',
In the case of '~는 것', his indicates the fact of doing something/being something. For example:
여기서 택시를 타기 쉬워요. (It is easy to 'take' a taxi here. (act=getting on a taxi)) 택시를 타는 것이 편해요. (It is convenient to take a taxi. You are not actually 'taking' the taxi, but explaining the fact that a taxi is convenient).
Hope this was of some help. Sincerely, Lyn Team KoreanClass101.com